On the antimagicness of generalized edge corona graphs

Abstract

Given a graph G, a function of assigning distinct labels {1,2,...,|E(G)|} to E(G) such that w(a)≠w(b), ∀ a,b∈V(G) is an antimagic labeling of G where w(a) indicates the vertex sum obtained by summing up all the labels assigned to the edges incident on the vertex a. Let G, Hi, 1≤i≤m be connected graphs such that E(G)={e1,e2,...,em}. A new graph is constructed from G, Hi, 1≤i≤m by adding all possible edges between the end vertices of ei and V(Hi), i∈{1,2,...,m}. The resulting graph is called the generalized edge corona of G and (H1,H2,...,Hm) which is denoted as G⋄(H1,H2,...,Hm). We prove G ⋄ (H1,H2,...,Hm) is antimagic under certain conditions using an algorithmic approach where G has only one vertex of maximum degree three (excluding spider graphs containing uneven legs) and |V(Hi)|≥2, i∈{1,2,...,m}.